There's Something About Gödel: The Complete Guide to the Incompleteness Theorem

by Francesco Berto (Ca' Foscari University of Venice, Italy )

• ISBN13: 9781444315011
• ISBN10: 1444315013
• Format: eBook
• Copyright: 2009-07-01
• Publisher: Wiley-Blackwell

Summary

This textbook explores the ways in which language informs the structure and function of the human mind, offering a point of entry into the fascinating territory of cognitive science. Focusing mainly on syntactic issues, Language in Cognition is a unique contribution to this burgeoning field of study. Guides undergraduate students through the core questions of linguistics and cognitive science, and provides tools that will help them think about the field in a structured way Uses the study of language and how language informs the structure and function of the human mind to introduce the major ideas in modern cognitive science, including its history and controversies Explores questions such as: what does it mean to say that linguistics is part of the cognitive sciences; how do the core properties of language compare with the core properties of other human cognitive abilities such as vision, music, mathematics, and other mental building blocks; and what is the relationship between language and thought? Includes an indispensable study guide as well as extensive references to encourage further independent study

Table of Contents

Prologue

Acknowledgments

The Gödelian Symphony

Foundations and Paradoxes

"This sentence is false"

The Liar and Gödel

Language and metalanguage

The axiomatic method, or how to get the non-obvious out of the obvious

Peano's axioms

... and the unsatisfied logicists, Frege and Russell

Bits of set theory

The Abstraction Principle

Bytes of set theory

Properties, relations, functions, that is, sets again

Calculating, computing, enumerating, that is, the notion of algorithm

Taking numbers as sets of sets

It's raining paradoxes

Cantor's diagonal argument

Self-reference and paradoxes

Hilbert

Strings of symbols

"... in mathematics there is no ignorabimus "

Gödel on stage

Our first encounter with the Incompleteness Theorem

... and some provisos

Gödelization, or Say It with Numbers!

TNT

The arithmetical axioms of TNT and the "standard model" N

The Fundamental Property of formal systems

The Gödel numbering

... and the arithmetization of syntax

Bits of Recursive Arithmetic

Making algorithms precise

Bits of recursion theory

Church's Thesis

The recursiveness of predicates, sets, properties, and relations

... And How It Is Represented in Typographical Number Theory

Introspection and representation

The representability of properties, relations, and functions

... and the Gödelian loop

"I Am Not Provable"

Proof pairs

The property of being a theorem of TNT (is not recursive!)

Arithmetizing substitution

How can a TNT sentence refer to itself?

¿

Fixed point

Consistency and omega-consistency

Proving G1

Rosser's proof

The Unprovability of Consistency and the "Immediate Consequences" of G1 and G2

G2

Technical interlude

"Immediate consequences" of G1 and G2

Undecidable1 and undecidable2

Essential incompleteness, or the syndicate of mathematicians

Robinson Arithmetic

How general are Gödel's results?

Bits of Turing machine

G1 and G2 in general

Unexpected fish in the formal net

Supernatural numbers

The culpability of the induction scheme

Bits of truth (not too much of it, though)

The World after Gödel

Bourgeois Mathematicians! The Postmodern Interpretations

What is postmodernism?

From Gödel to Lenin

Is "Biblical proof" decidable?

Speaking of the totality

Bourgeois teachers!

(Un)interesting bifurcations

A Footnote to Plato

Explorers in the realm of numbers

The essence of a life

"The philosophical prejudices of our times"

From Gödel to Tarski

Human, too human

Mathematical Faith

"I'm not crazy!"

Qualified doubts

From Gentzen to the Dialectica interpretation

Mathematicians are people of faith

Mind versus Computer: Gödel and Artificial Intelligence

Is mind (just) a program?

"Seeing the truth" and "going outside the system"

The basic mistake

In the haze of the transfinite

"Know thyself": Socrates and the inexhaustibility of mathematics

Gödel versus Wittgenstein and the Paraconsistent Interpretation

When geniuses meet

The implausible Wittgenstein

"There is no metamathematics"

Proof and prose

The single argument

But how can arithmetic be inconsistent?

The costs and benefits of making Wittgenstein plausible

Epilogue

References

Index

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